There is an error in the second step. Before cross multiplying the denominators, LCM of the values should be taken.
Step-by-step explanation:
Step 1; 1 - [tex]\frac{2}{x-2}[/tex] = [tex]\frac{x+ 1}{x+ 2}[/tex].
By taking an LCM of x-2 on the LHS, we get
[tex]\frac{x-2}{x-2}[/tex] - [tex]\frac{2}{x-2}[/tex] = [tex]\frac{x+ 1}{x+ 2}[/tex],
as both terms on the LHS have same denominator, we can add them up,
[tex]\frac{x-2-2}{x-2}[/tex] = [tex]\frac{x-4}{x-2}[/tex] = [tex]\frac{x+ 1}{x+ 2}[/tex].
Step 2; Now we cross multiply the denominators.
(x - 4) × (x + 2) = (x + 1) × (x - 2),
x² - 2x - 8 = x² - x - 2,
(x² - 2x - 8) - (x² - x - 2) = 0.
x² - x² - 2x + x - 8 + 2 = 0,
-x -6 = 0,
x = -6.
By simplifying the given expression, we get x = -6.
